I didn't see much difference between the movies and it's been about two years since I read the books for only the second time. Gimli says in the movie, "This new Gandalf is more grumpy than the old one." But that's the only overt observation in words or behavior that I remember from the movies.

Physically White Gandalf is taller, more imposing with a more noble image than the Pilgrim of Grey. His face is softer and the hook nose looks smaller. Ian McKellen speaks with a lighter, more benevolent tone for White instead of the deeper, more gravelly voice of Grey. Personality wise he appears wiser and more resolute but this makes him come across as more serious and less fun than Grey which is probably why most people including McKellen prefer the former incarnation. "A single dream is more powerful than a thousand realities."